Joint use of time-domain and windowed method for design of modulated DFT filter banks

A novel and efficient technique to design modulated discrete Fourier transform （DFT） filter banks is introduced in this paper. The proposed method first relaxes the limits that the synthesis filters are the time-reversed version of the analysis filters and then adopts the time domain formula of the perfect reconstruction property as the solution to design the synthesis filters. The prototype filter in analysis filter banks is designed based on Fourier-Kaiser window approach. Simulation results show that the designed filter banks approximately satisfy the perfect reconstruction with controllable reconstruction errors.     

M-ary wavelet transform and formulation for perfect reconstructionin M-band filter bank

The binary wavelet transform is generalized and extended to the M-ary biorthonormal case. The computational equivalence between the discrete wavelet analysis and the M-band multirate signal filtering is indicated. The equivalence allows the perfect reconstruction requirement in a filter bank to be investigated from the vector space decomposition/reconstruction in wavelet analysis. From the construction of the biorthonormal wavelet bases, the necessary and sufficient condition for the filters in a perfect reconstruction filter bank is formulated. Under this formulation, an additional optimization procedure is then used to model the frequency domain requirement in filter bank design     

Analog wavelet transform using multiple-loop feedback switched-current filters and simulated annealing algorithms

A new approach for implementing continuous wavelet transform (CWT) based on multiple-loop feedback (MLF) switched-current (SI) filters and simulated annealing algorithms (SAA) is presented. First, the approximation function of wavelet bases is performed by employing SAA. This approach allows for the circuit implementation of any other wavelets. Then the wavelet filter whose impulse response is the wavelet approximation function is designed using MLF architectures, which is constructed with SI differentiators and multi-output cascade current source circuits. Finally, the CWT is implemented by controlling the clock frequency of wavelet filter banks. Simulation results of the proposed circuits and the filter banks show the advantages of such new designs.     

Filters and filter banks for periodic signals, the Zak transform,and fast wavelet decomposition

We present a new approach to filtering and reconstruction of periodic signals. The tool that proves to handle these tasks very efficiently is the discrete Zak transform. The discrete Zak transform can be viewed as the discrete Fourier transform performed on the signal blocks. It also can be considered the polyphase representation of periodic signals. Fast filtering-decimation-interpolation-reconstruction algorithms are developed in the Zak transform domain both for the undersampling and critical sampling cases. The technique of finding the optimal biorthogonal filter banks, i.e., those that would provide the best reconstruction even in the undersampling case, is presented. An algorithm for orthogonalization of nonorthogonal filters is developed. The condition for perfect reconstruction for the periodic signals is derived. The generalizations are made for the nonperiodic sequences, and several ways to apply the developed technique to the nonperiodic sequences are considered. Finally, the developed technique is applied to recursive filter banks and the discrete wavelet decomposition     

A binary wavelet decomposition of binary images

We construct a theory of binary wavelet decompositions of finite binary images. The new binary wavelet transform uses simple module-2 operations. It shares many of the important characteristics of the real wavelet transform. In particular, it yields an output similar to the thresholded output of a real wavelet transform operating on the underlying binary image. We begin by introducing a new binary field transform to use as an alternative to the discrete Fourier transform over GF(2). The corresponding concept of sequence spectra over GF(2) is defined. Using this transform, a theory of binary wavelets is developed in terms of two-band perfect reconstruction filter banks in GF(2). By generalizing the corresponding real field constraints of bandwidth, vanishing moments, and spectral content in the filters, we construct a perfect reconstruction wavelet decomposition. We also demonstrate the potential use of the binary wavelet decomposition in lossless image coding.     

A new class of two-channel biorthogonal filter banks and waveletbases

Proposes a novel framework for a new class of two-channel biorthogonal filter banks. The framework covers two useful subclasses: i) causal stable IIR filter banks. ii) linear phase FIR filter banks. There exists a very efficient structurally perfect reconstruction implementation for such a class. Filter banks of high frequency selectivity can be achieved by using the proposed framework with low complexity. The properties of such a class are discussed in detail. The design of the analysis/synthesis systems reduces to the design of a single transfer function. Very simple design methods are given both for FIR and IIR cases. Zeros of arbitrary multiplicity at aliasing frequency can be easily imposed, for the purpose of generating wavelets with regularity property. In the IIR case, two new classes of IIR maximally flat filters different from Butterworth filters are introduced. The filter coefficients are given in closed form. The wavelet bases corresponding to the biorthogonal systems are generated. the authors also provide a novel mapping of the proposed 1-D framework into 2-D. The mapping preserves the following: i) perfect reconstruction; ii) stability in the IIR case; iii) linear phase in the FIR case; iv) zeros at aliasing frequency; v) frequency characteristic of the filters     

用于信号逼近的最佳正交子波选择方法

离散子波变换将离散时间信号分解为一系列分辨率下的离散逼近和离散细节。紧支的正交规范子波与完全重建正交镜象滤波器组相对应。本文提出一种用于信号最佳逼近的正交子波选择方法,即选择满足一定条件的滤波器的方法。通过对滤波器参数化,可以将带约束的最优化问题转化为无约束最优化问题,通过对参数在一定范围内的搜索,得到最优解。文中给出了计算机模拟的结果。     

A wavelet transform with point-symmetric extension at tile boundaries

This paper presents a low-complexity wavelet transform that utilizes point-symmetric extension at the image tile boundaries. The proposed approach is considered as an alternative to the well-known symmetric signal extension in dealing with the boundary artifacts that manifest when tiles of images are lossy compressed. The new solution developed for odd-length filters preserves the perfect reconstruction property of the filter banks and deals efficiently with blocking artifacts without requiring any post processing technique or additional information from the neighboring tiles. It is shown that point-symmetric extension at the tile boundaries does not need to be applied explicitly, but instead the equivalent boundary filters can be derived. The lifting-based implementation of the filters provides a very simple way of changing filter parameters at the boundaries that suits both hardware and software platforms.     

用余弦调制PR－FIR方法构造正交小波基

离散小波变换将离散时间信号分解为一系列不同分辨率下的离散近似信号和离散细节，紧支的正交规范小波与完全重构正交镜象滤波器（ＰＲ－ＱＭＦ）相对应。本文在“二带”正交小波基的构造条件下，利用余弦调制完全重构滤波器组的方法，实现了正交小波基的构造，计算模拟表明该方法非常简单、有效。     

Multidimensional Two-channel Linear Phase FIR Filter Banks and Wavelet Bases with Vanishing Moments

In the paper new design methods for multidimensional two-channel perfect reconstruction (PR) FIR filter banks are described. The filters have linear phase, achieve an arbitrarily high number of vanishing moments and can be used to obtain biorthogonal wavelet bases. Two-dimensional wavelet bases with square, rectangular, and hexagonal support are constructed. A design method that trades off vanishing moments and frequency selectivity is described. These two properties are both important in subband coding, but they each require a share of the degrees of freedom available from the total number of filter coefficients. Novel analytic formulae are derived and are applied to provide illustrative examples.     

Generalized Chebyshev Filters for the Design of IIR Filters and Filter Banks

This paper introduces the generalized IIR Chebyshev filters. The proposed filters are obtained by applying bilinear transformation to the corresponding analog filters. The novelty of the method is the introduction of a new rational Chebyshev function, which includes Chebyshev Type I and Chebyshev Type II IIR filters as special cases. The application of the proposed digital filters to design perfect reconstruction two-channel filter banks is described. The proposed filters can be applied in orthogonal discrete wavelet transform.     

连续小波变换开关电流电路的实现

提出了一种用开关电流电路实现连续小波变换的方法,将连续小波变换转化为用带通滤波器组对信号进行处理,并用开关电流电路实现该带通滤波器组.文章采用基于第二代开关电流技术的带通滤波器组实现了8通道的Marr小波.仿真结果表明该滤波器组具有恒Q值,且每个带通滤波器的中心频率与理论值大致相符,从而证实了该方法的可行性.     

波形匹配小波设计及其在信号数据压缩中的应用

小波变换被广泛应用于信号数据压缩领域,如何选择小波基,达到用稀疏系数最优表示信号是提高压缩比的关键问题之一。该文提出了选用与信号波形最优相似的小波基压缩一维信号,并提出了构造此类小波基的方法。利用结构化小波滤波器组以及新的波形相似准则,给出了构造最优匹配小波基函数的算法。将此小波应用于一维信号数据压缩,结果表明,与其它的小波相比,所构造的最优匹配小波对信号具有更好的压缩性能,说明文中提出的方法是有效可行的。     

Technique for design of two-channel approximately linear phase QMFfilter bank and its application to image compression

The two-channel QMF filter bank based on allpass sections is one of the best known circuits for building up a multi-channel filter bank for signal compression. An analysis-synthesis combination can satisfy two of the three perfect reconstruction (PR) conditions. The third, the phase condition, can be met to any desired accuracy. However, when applied to images, PR is possible because non-causality can be allowed in the synthesis bank. The development of explicit formulae for the coefficients of such filters is considered. The resulting QMF designs can be used in a wavelet structure for image compression problems using rectangular or diamond symmetry. Several design examples are given, including a comparison of performance with FIR filter banks for such problems     

Pade table, continued fraction expansion, and perfectreconstruction filter banks

We investigate the relationships among the Pade table, continued fraction expansions and perfect reconstruction (PR) filter banks. We show how the Pade table can be utilized to develop a new lattice structure for general two-channel bi-orthogonal perfect reconstruction (PR) filter banks. This is achieved through characterization of all two-channel biorthogonal PR filter banks. The parameterization found using this method is unique for each filter bank. Similar to any other lattice structure, the PR property is achieved structurally and the quantization of the parameters of the lattice does not effect this property. Furthermore, we demonstrate that for a given filter, the set of all complementary filters can be uniquely specified by two parameters, namely, the end-to-end delay of the system and a scalar quantity. Finally, we investigate the convergence of the successive filters found through the proposed lattice structure and develop a sufficient condition for this convergence     

CONSTRUCTION METHODS AND ALGORITHM DESIGN OF WAVELET BASES

Based on the brief introduction of the principles of wavelet analysis, this paper gives a summary of several typical wavelet bases from the point of view of perfect reconstruction of signals and emphasizes that designing wavelet bases which are used to decompose the signal into a two-band form is equivalent to designing a two-band filter bank with perfect or nearly perfect property. The generating algorithm corresponding to Daubechies bases and some simulated results are also given in the paper.     

A general formulation of modulated filter banks

This paper presents a general framework for maximally decimated modulated filter banks. The theory covers the known classes of cosine modulation and relates them to complex-modulated filter banks. The prototype filters have arbitrary lengths, and the overall delay of the filter bank is arbitrary, within fundamental limits. Necessary and sufficient conditions for perfect reconstruction (PR) are derived using the polyphase representation. It is shown that these PR conditions are identical for all types of modulation-modulation based on the discrete cosine transform (DCT), both DCT-III/DCT-IV and DCT-I/DCT-II, and modulation based on the modified discrete Fourier transform (MDFT). A quadratic-constrained design method for prototype filters yielding PR with arbitrary length and system delay is derived, and design examples are presented to illustrate the tradeoff between overall system delay and stopband attenuation (subchannelization)     

基于加权总体最小二乘正则化方法的混合滤波器组最优化设计

模拟分析滤波器组的实现欠理想、系统噪声以及数字综合滤波器有效阶数实现所带来的系统误差均有可能造成混合滤波器组的设计出现解不稳定、无唯一解等病态问题,影响混合滤波器组的准确重构效果。本文首先给出了满足准确重构条件下,以综合滤波器组频域响应为求解变量的混合滤波器组线性求解模型。针对线性方程中系数矩阵以及目标向量受扰动误差影响特点,提出一种新的基于加权总体最小二乘正则化算法的IIR形式综合滤波器设计方法。算法以系统扰动误差最小化为目标函数,根据随机误差变量的二阶统计特性,采用加权总体最小二乘算法抑制滤波器实现误差以及随机噪声等扰动因素影响,使得到的综合滤波器组频域响应解的加权误差平方和最小化,并通过Tikhonov正则化方法优化病态情况下方程组解的稳定性。提出一种IIR类型的综合滤波器系数的求解算法,并利用正则化方法优化滤波器系数,提高系统稳定性。该方法可应用于过采样混合滤波器组的设计。仿真结果表明该算法的有效提高系统鲁棒性和改善重构性能。      

Multidimensional multirate filters and filter banks derived fromone-dimensional filters

A method by which every multidimensional (M-D) filter with an arbitrary parallelepiped-shaped passband support can be designed and implemented efficiently is presented. It is shown that all such filters can be designed starting from an appropriate one-dimensional prototype filter and performing a simple transformation. With D denoting the number of dimensions, the complexity of design and implementation of the M-D filter are reduced from O(N^D) to O(N). Using the polyphase technique, an implementation with complexity of only 2N is obtained in the two-dimensional. Even though the filters designed are in general nonseparable, they have separable polyphase components. One special application of this method is in M-D multirate signal processing, where filters with parallelepiped-shaped passbands are used in decimation, interpolation, and filter banks. Some generalizations and other applications of this approach, including M-D uniform discrete Fourier transform (DFT) quadrature mirror filter banks that achieve perfect reconstruction, are studied. Several design example are given     

A Novel Scheme for the Design of Approximate Hilbert Transform Pairs of Orthonormal Wavelet Bases

In designing the Hilbert transform pairs of orthonormal wavelet bases, several authors have shown that the requirements of the equal magnitude responses and the half-sample phase offset are the necessary and sufficient conditions on the scaling filters of conjugate quadrature filter (CQF) banks. In this paper, we first show that the scaling filters of CQF banks with equal magnitude responses cannot approach the half-sample phase offset in the high-frequency range properly. Then, a new design scheme, making a tradeoff between the requirements of the equal magnitude responses and the half-sample phase offset, is presented. The design scheme can make the scaling filter pairs approximately satisfy the requirement of the half-sample phase offset over the full frequency range while the magnitude responses of each pair remain nearly equal. As a result, the corresponding orthonormal wavelet bases can approximate to Hilbert transform pairs better. Some orthogonal wavelet bases we designed demonstrate the superiority of our scheme.